Difficult ODE with Heaviside as coefficient So In my research I stumbled upon a difficult ODE,
It goes like this
$$ y''(x)-[(\operatorname{Heaviside}(ax)+b]y(x)=0, $$ (a,b are the respective constants)
I tried approximating Heaviside with analytical terms, which gave me these ODE's
$$ y''(x)-[b/2*(1+2/π * \arctan⁡(x/ϵ) )+a]y(x)=0,  $$
or
$$ y''(x)-[a/(1+e^x\epsilon]+b)y(x)=0$$
The epsilon should multiply the power as well, it doesn't work for some reason.
The mighty wolfram gives me answers that I barely understand,
How would you suggest I go about it? Should I try a different approach? is it even solvable?
Laplace transform also didn't help me.
Thanks a lot!
 A: Assumptions. $a$ is positive and initial conditions $y(x_{0})$ and $y^{\prime}(x_{0})$ are specified at $x_{0}<0$.
As John Barber points out, $H(ax)=H(x)$ in this case.
Therefore, the ODE is equivalent to
$$
y^{\prime\prime}(x)=\begin{cases}
\left(b+0\right)y(x) & \text{if }x<0\\
\left(b+1\right)y(x) & \text{if }x>0.
\end{cases}
$$
It follows that
$$
y(x)=c_{1}e^{x\sqrt{b}}+c_{2}e^{-x\sqrt{b}}\qquad\text{for }x_{0}<x\leq0
$$
where $c_{1}$ and $c_{2}$ are obtained by solving the linear system
$$
\begin{pmatrix}\phantom{\sqrt{b}}e^{x_{0}\sqrt{b}} & \phantom{-\sqrt{b}}e^{-x_{0}\sqrt{b}}\\
\sqrt{b}e^{x_{0}\sqrt{b}} & -\sqrt{b}e^{-x_{0}\sqrt{b}}
\end{pmatrix}\begin{pmatrix}c_{1}\\
c_{2}
\end{pmatrix}=\begin{pmatrix}y(x_{0})\\
y^{\prime}(x_{0})
\end{pmatrix}.
$$
Similarly,
$$
y(x)=C_{1}e^{x\sqrt{b+1}}+C_{2}e^{-x\sqrt{b+1}}\qquad\text{for }x>0
$$
where $C_{1}$ and $C_{2}$ are obtained by solving the linear system
$$
\begin{pmatrix}1 & 1\\
\sqrt{b+1} & -\sqrt{b+1}
\end{pmatrix}\begin{pmatrix}C_{1}\\
C_{2}
\end{pmatrix}=\begin{pmatrix}y(0)\\
y^{\prime}(0)
\end{pmatrix}.
$$
